The paradox of knowability is a logical result suggesting that, necessarily, if all truths are knowable in principle then all truths are in fact known. The contrapositive of the result says, necessarily, if in fact there is an unknown truth, then there is a truth that couldn’t possibly be known. More specifically, if p is a truth that is never known then it is unknowable that p is a truth that is never known.

Some believe that everything in the universe can be known through the process of science. Everything that is not known today will be known eventually. I do not quite agree. Some things are, I think, unfathomable, unthinkable – which one can not know. My reasoning is quite simple. The capacity of our brains is limited. That seems undisputed. It is because of our cognitive limitations, we have found it necessary in our invented languages to invent the concept of infinity – for things that are beyond observable. If brain capacity was unlimited there would be no need for the concept of infinity or for words such as “incomprehensible”. Even in mathematics, which, after all, is another language for describing the universe, we also have the concept of infinity. Infinitely large and infinitely small. We can find finite limits for some convergent infinite series, but we can never get to infinity by making finite operations on finite numbers. We cannot fill an infinite volume with a finite bucket or measure an infinitely long line with a finite ruler.

even our most fundamental axioms in philosophy and mathematics are assumed “true” but cannot be proven to be so, and

there are areas of Kant’s “noumena” which are not capable of being known

It is not just that we do not know what we do not know. We cannot know anything of what is unknowable. In the universe of the knowable, the unknowable lies in the black holes of the current universe and in the time before time began.

Noumenon, (plural Noumena), in the philosophy of Immanuel Kant, the thing-in-itself (das Ding an sich) as opposed to what Kant called the phenomenon—the thing as it appears to an observer. Though the noumenal holds the contents of the intelligible world, Kant claimed that man’s speculative reason can only know phenomena and can never penetrate to the noumenon.

All human logic and all human reasoning are based on the assumption that what is not true is false, and what is not false must be true. But why is it that there cannot be a state where something is neither true nor false? Why not both false and true at the same time? In fact, it is the logic inherent in our own language that prohibits this state. It is a limitation of language which we cannot avoid. But language is not discovered, it is invented. That “what is not true must be false” is just an assumption – a very basic and deep assumption but still just an assumption.

Following Fitch

in fact there are unknown truths, therefore there must be truths that couldn’t possibly be known.

Therefore,

“Truth” and “that which is not true is false” are assumptions and are definitions inbuilt in our languages. Truth and Falsity are not necessarily mutually exclusive quantum states. They may instead form part of a continuum which is unknowable. Maybe a truth-time continuum?